Delaunay triangulations are widely used in scientific computing in many diverse applications. While there are numerous algorithms for computing triangulations, it is the favorable geometric properties of the Delaunay triangulation that make it so useful.

The fundamental property is the Delaunay criterion. In the case
of 2-D triangulations, this is often called the empty circumcircle
criterion. For a set of points in 2-D, a Delaunay triangulation of
these points ensures the circumcircle associated with each triangle
contains no other point in its interior. This property is important.
In the illustration below, the circumcircle associated with `T1`

is
empty. It does not contain a point in its interior. The circumcircle
associated with `T2`

is empty. It does not contain
a point in its interior. This triangulation is a Delaunay triangulation.

The triangles below are different. The circumcircle associated
with `T1`

is not empty. It contains `V3`

in
its interior. The circumcircle associated with `T2`

is
not empty. It contains `V1`

in its interior. This
triangulation is *not* a Delaunay triangulation.

Delaunay triangles are said to be “well shaped”
because in fulfilling the empty circumcircle property, triangles with
large internal angles are selected over ones with small internal angles.
The triangles in the non-Delaunay triangulation have sharp angles
at vertices `V2`

and `V4`

. If the
edge `{V2, V4}`

were
replaced by an edge joining `V1`

and `V3`

,
the minimum angle would be maximized and the triangulation would become
a Delaunay triangulation. Also, the Delaunay triangulation connects
points in a nearest-neighbor manner. These two characteristics, well-shaped
triangles and the nearest-neighbor relation, have important implications
in practice and motivate the use of Delaunay triangulations in scattered
data interpolation.

While the Delaunay property is well defined, the topology of the triangulation is not unique in the presence of degenerate point sets. In two dimensions, degeneracies arise when four or more unique points lie on the same circle. The vertices of a square, for example, have a nonunique Delaunay triangulation.

The properties of Delaunay triangulations extend to higher dimensions. The triangulation of a 3-D set of points is composed of tetrahedra. The next illustration shows a simple 3-D Delaunay triangulation made up of two tetrahedra. The circumsphere of one tetrahedron is shown to highlight the empty circumsphere criterion.

A 3-D Delaunay triangulation produces tetrahedra that satisfy the empty circumsphere criterion.

MATLAB^{®} provides two ways to create Delaunay triangulations:

The functions

`delaunay`

and`delaunayn`

The

`delaunayTriangulation`

class

The `delaunay`

function supports the creation
of 2-D and 3-D Delaunay triangulations. The `delaunayn`

function
supports creating Delaunay triangulations in 4-D and higher.

Creating Delaunay triangulations in dimensions higher than 6-D is generally not practical for moderate to large point sets due to the exponential growth in required memory.

The `delaunayTriangulation`

class
supports creating Delaunay triangulations in 2-D and 3-D. It provides
many methods that are useful for developing triangulation-based algorithms.
These class methods are like functions, but they are restricted to
work with triangulations created using `delaunayTriangulation`

. The `delaunayTriangulation`

class
also supports the creation of related constructs such as the convex
hull and Voronoi diagram. It also supports the creation of constrained
Delaunay triangulations.

In summary:

The

`delaunay`

function is useful when you only require the basic triangulation data, and that data is sufficiently complete for your application.The

`delaunayTriangulation`

class offers more functionality for developing triangulation-based applications. It is useful when you require the triangulation and you want to perform any of these operations:Search the triangulation for triangles or tetrahedra enclosing a query point.

Use the triangulation to perform a nearest-neighbor point search.

Query the triangulation's topological adjacency or geometric properties.

Modify the triangulation to insert or remove points.

Constrain edges in the triangulation—this is called a constrained Delaunay triangulation.

Triangulate a polygon and optionally remove the triangles that are outside of the domain.

Use the Delaunay triangulation to compute the convex hull or Voronoi diagram.

The `delaunay`

and `delaunayn`

functions take a set of points
and produce a triangulation in matrix format. Refer to Triangulation Matrix Format for more
information on this data structure. In 2-D, the `delaunay`

function is often used to produce
a triangulation that can be used to plot a surface defined in terms
of a set of scattered data points. In this application, it’s
important to note that this approach can only be used if the surface
is single-valued. For example, it could not be used to plot a spherical
surface because there are two `z`

values corresponding
to a single (`x`

, `y`

) coordinate.
A simple example demonstrates how the `delaunay`

function
can be used to plot a surface representing a sampled data set.

This example shows how to use the `delaunay`

function to create a 2-D Delaunay triangulation from the seamount data set. A *seamount* is an underwater mountain. The data set consists of a set of longitude (`x`

) and latitude (`y`

) locations, and corresponding seamount elevations (`z`

) measured at those coordinates.

Load the seamount data set and view the (`x`

, `y`

) data as a scatter plot.

load seamount plot(x,y,'.','markersize',12) xlabel('Longitude'), ylabel('Latitude') grid on

Construct a Delaunay triangulation from this point set and use `triplot`

to plot the triangulation in the existing figure.

tri = delaunay(x,y); hold on, triplot(tri,x,y), hold off

Add the depth data (`z`

) from seamount to lift the vertices and create the surface. Create a new figure and use `trimesh`

to plot the surface in wireframe mode.

figure hidden on trimesh(tri,x,y,z) xlabel('Longitude'),ylabel('Latitude'),zlabel('Depth in Feet');

If you want to plot the surface in shaded mode, use `trisurf`

instead of `trimesh`

.

A 3-D Delaunay triangulation also can be created using the `delaunay`

function. This triangulation is composed of tetrahedra.

This example shows how to create a 3-D Delaunay triangulation of a random data set. The triangulation is plotted using `tetramesh`

, and the `FaceAlpha`

option adds transparency to the plot.

X = gallery('uniformdata',[30 3],0); tet = delaunay(X); faceColor = [0.6875 0.8750 0.8984]; tetramesh(tet,X,'FaceColor', faceColor,'FaceAlpha',0.3);

MATLAB provides the `delaunayn`

function
to support the creation of Delaunay triangulations in dimension 4-D
and higher. Two complementary functions `tsearchn`

and `dsearchn`

are also provided to support
spatial searching for N-D triangulations. See Spatial Searching for more information on triangulation-based
search.

The `delaunayTriangulation`

class
provides another way to create Delaunay triangulations in MATLAB.
While `delaunay`

and `delaunayTriangulation`

use the
same underlying algorithm and produce the same triangulation, `delaunayTriangulation`

provides
complementary methods that are useful for developing Delaunay-based
algorithms. These methods are like functions that are packaged together
with the triangulation data into a container called a class. Keeping
everything together in a class provides a more organized setup that
improves ease of use. It also improves the performance of triangulation-based
searches such as point-location and nearest-neighbor. `delaunayTriangulation`

supports
incremental editing of the Delaunay triangulation. You also can impose
edge constraints in 2-D.

Triangulation Representations introduces
the `triangulation`

class,
which supports topological and geometric queries for 2-D and 3-D triangulations.
A `delaunayTriangulation`

is
a special kind of `triangulation`

.
This means you can perform any `triangulation`

query
on a `delaunayTriangulation`

in
addition to the Delaunay-specific queries. In more formal MATLAB language
terms, `delaunayTriangulation`

is
a subclass of `triangulation`

.

This example shows how to create, query, and edit a Delaunay triangulation from the `seamount`

data using `delaunayTriangulation`

. The seamount data set contains (`x`

, `y`

) locations and corresponding elevations (`z`

) that define the surface of the seamount.

Load and triangulate the (`x`

, `y`

) data.

```
load seamount
DT = delaunayTriangulation(x,y)
```

DT = delaunayTriangulation with properties: Points: [294x2 double] ConnectivityList: [566x3 double] Constraints: []

The `Constraints`

property is empty because there aren't any imposed edge constraints. The `Points`

property represents the coordinates of the vertices, and the `ConnectivityList`

property represents the triangles. Together, these two properties define the matrix data for the triangulation.

The `delaunayTriangulation`

class is a wrapper around the matrix data, and it offers a set of complementary methods. You access the properties in a `delaunayTriangulation`

in the same way you access the fields of a struct.

Access the vertex data.

DT.Points;

Access the connectivity data.

DT.ConnectivityList;

Access the first triangle in the `ConnectivityList`

property.

DT.ConnectivityList(1,:)

`ans = `*1×3*
205 230 262

`delaunayTriangulation`

provides an easy way to index into the `ConnectivityList`

property matrix.

Access the first triangle.

DT(1,:)

`ans = `*1×3*
205 230 262

Examine the first vertex of the first triangle.

DT(1,1)

ans = 205

Examine all the triangles in the triangulation.

DT(:,:);

Indexing into the `delaunayTriangulation`

output, `DT`

, works like indexing into the triangulation array output from `delaunay`

. The difference between the two are the extra methods that you can call on `DT`

(for example, `nearestNeighbor`

and `pointLocation`

).

Use `triplot`

to plot the `delaunayTriangulation`

. The `triplot`

function is not a `delaunayTriangulation`

method, but it accepts and can plot a `delaunayTriangulation`

.

triplot(DT); axis equal xlabel('Longitude'), ylabel('Latitude') grid on

Alternatively, you could use `triplot(DT(:,:), DT.Points(:,1), DT.Points(:,2));`

to get the same plot.

Use the `delaunayTriangulation`

method, `convexHull`

, to compute the convex hull and add it to the plot. Since you already have a Delaunay triangulation, this method allows you to derive the convex hull more efficiently than a full computation using `convhull`

.

hold on k = convexHull(DT); xHull = DT.Points(k,1); yHull = DT.Points(k,2); plot(xHull,yHull,'r','LineWidth',2); hold off

You can incrementally edit the `delaunayTriangulation`

to add or remove points. If you need to add points to an existing triangulation, then an incremental addition is faster than a complete retriangulation of the augmented point set. Incremental removal of points is more efficient when the number of points to be removed is small relative to the existing number of points.

Edit the triangulation to remove the points on the convex hull from the previous computation.

figure plot(xHull,yHull,'r','LineWidth',2); axis equal xlabel('Longitude'),ylabel('Latitude') grid on % The convex hull topology duplicates the start and end vertex. % Remove the duplicate entry. k(end) = []; % Now remove the points on the convex hull. DT.Points(k,:) = []

DT = delaunayTriangulation with properties: Points: [274x2 double] ConnectivityList: [528x3 double] Constraints: []

% Plot the new triangulation. hold on triplot(DT); hold off

There is one vertex that is just inside the boundary of the convex hull that was not removed. The fact that it is interior to the hull can be seen using the Zoom-In tool in the figure. You could plot the vertex labels to determine the index of this vertex and remove it from the triangulation. Alternatively, you can use the `nearestNeighbor`

method to identify the index more readily.

The point is close to location (211.6, -48.15). Use the nearestNeighbor method to find the nearest vertex.

vertexId = nearestNeighbor(DT, 211.6, -48.15)

vertexId = 50

Now remove that vertex from the triangulation.

DT.Points(vertexId,:) = []

DT = delaunayTriangulation with properties: Points: [273x2 double] ConnectivityList: [525x3 double] Constraints: []

Plot the new triangulation.

figure plot(xHull,yHull,'r','LineWidth',2); axis equal xlabel('Longitude'),ylabel('Latitude') grid on hold on triplot(DT); hold off

Add points to the existing triangulation. Add 4 points to form a rectangle around the triangulation.

```
Padditional = [210.9 -48.5; 211.6 -48.5; ...
211.6 -47.9; 210.9 -47.9];
DT.Points(end+(1:4),:) = Padditional
```

DT = delaunayTriangulation with properties: Points: [277x2 double] ConnectivityList: [548x3 double] Constraints: []

Close all existing figures.

`close all`

Plot the new triangulation.

figure plot(xHull,yHull,'r','LineWidth',2); axis equal xlabel('Longitude'),ylabel('Latitude') grid on hold on triplot(DT); hold off

You can edit the points in the triangulation to move them to a new location. Edit the first of the additional point set (the vertex ID 274).

DT.Points(274,:) = [211 -48.4];

Close all existing figures.

`close all`

Plot the new triangulation

figure plot(xHull,yHull,'r','LineWidth',2); axis equal xlabel('Longitude'),ylabel('Latitude') grid on hold on triplot(DT); hold off

Use the a method of the `triangulation`

class, `vertexAttachments`

, to find the attached triangles. Since the number of triangles attached to a vertex is variable, the method returns the attached triangle IDs in a cell array. You need braces to extract the contents.

attTris = vertexAttachments(DT,274); hold on triplot(DT(attTris{:},:),DT.Points(:,1),DT.Points(:,2),'g') hold off

`delaunayTriangulation`

also can be used to triangulate points in 3-D space. The resulting triangulation is composed of tetrahedra.

This example shows how to use a `delaunayTriangulation`

to create and plot the triangulation of 3-D points.

```
P = gallery('uniformdata',30,3,0);
DT = delaunayTriangulation(P)
```

DT = delaunayTriangulation with properties: Points: [30x3 double] ConnectivityList: [117x4 double] Constraints: []

faceColor = [0.6875 0.8750 0.8984]; tetramesh(DT,'FaceColor', faceColor,'FaceAlpha',0.3);

The `tetramesh`

function plots both the internal and external faces of the triangulation. For large 3-D triangulations, plotting the internal faces might be an unnecessary use of resources. A plot of the boundary might be more appropriate. You can use the `freeBoundary`

method to get the boundary triangulation in matrix format. Then pass the result to `trimesh`

or `trisurf`

.

The `delaunayTriangulation`

class
allows you to constrain edges in a 2-D triangulation. This means you
can choose a pair of points in the triangulation and constrain an
edge to join those points. You can picture this as “forcing”
an edge between one or more pairs of points. The following example
shows how edge constraints can affect the triangulation.

The triangulation below is a Delaunay triangulation because it respects the empty circumcircle criterion.

Triangulate a set of points with an edge constraint specified between vertex `V1`

and `V3`

.

Define the point set.

P = [2 4; 6 1; 9 4; 6 7];

Define a constraint, `C`

, between `V1`

and `V3`

.

C = [1 3]; DT = delaunayTriangulation(P,C);

Plot the triangulation and add annotations.

triplot(DT) % Label the vertices. hold on numvx = size(P,1); vxlabels = arrayfun(@(n) {sprintf('V%d', n)}, (1:numvx)'); Hpl = text(P(:,1)+0.2, P(:,2)+0.2, vxlabels, 'FontWeight', ... 'bold', 'HorizontalAlignment','center', 'BackgroundColor', ... 'none'); hold off % Use the incenters to find the positions for placing triangle labels on the plot. hold on IC = incenter(DT); numtri = size(DT,1); trilabels = arrayfun(@(P) {sprintf('T%d', P)}, (1:numtri)'); Htl = text(IC(:,1),IC(:,2),trilabels,'FontWeight','bold', ... 'HorizontalAlignment','center','Color','blue'); hold off % Plot the circumcircle associated with the triangle, T1. hold on [CC,r] = circumcenter(DT); theta = 0:pi/50:2*pi; xunit = r(1)*cos(theta) + CC(1,1); yunit = r(1)*sin(theta) + CC(1,2); plot(xunit,yunit,'g'); axis equal hold off

The constraint between vertices (`V1`

, `V3`

) was honored, however, the Delaunay criterion was invalidated. This also invalidates the nearest-neighbor relation that is inherent in a Delaunay triangulation. This means the `nearestNeighbor`

search method provided by `delaunayTriangulation`

cannot be supported if the triangulation has constraints.

In typical applications, the triangulation might be composed of many points, and a relatively small number of edges in the triangulation might be constrained. Such a triangulation is said to be locally non-Delaunay, because many triangles in the triangulation might respect the Delaunay criterion, but locally there might be some triangles that do not. In many applications, local relaxation of the empty circumcircle property is not a concern.

Constrained triangulations are generally used to triangulate a nonconvex polygon. The constraints give us a correspondence between the polygon edges and the triangulation edges. This relationship enables you to extract a triangulation that represents the region. The following example shows how to use a constrained `delaunayTriangulation`

to triangulate a nonconvex polygon.

Define and plot a polygon.

figure() axis([-1 17 -1 6]); axis equal P = [0 0; 16 0; 16 2; 2 2; 2 3; 8 3; 8 5; 0 5]; patch(P(:,1),P(:,2),'-r','LineWidth',2,'FaceColor',... 'none','EdgeColor','r'); % Label the points. hold on numvx = size(P,1); vxlabels = arrayfun(@(n) {sprintf('P%d', n)}, (1:numvx)'); Hpl = text(P(:,1)+0.2, P(:,2)+0.2, vxlabels, 'FontWeight', ... 'bold', 'HorizontalAlignment','center', 'BackgroundColor', ... 'none'); hold off

Create and plot the triangulation together with the polygon boundary.

figure() subplot(2,1,1); axis([-1 17 -1 6]); axis equal P = [0 0; 16 0; 16 2; 2 2; 2 3; 8 3; 8 5; 0 5]; DT = delaunayTriangulation(P); triplot(DT) hold on; patch(P(:,1),P(:,2),'-r','LineWidth',2,'FaceColor',... 'none','EdgeColor','r'); hold off % Plot the standalone triangulation in a subplot. subplot(2,1,2); axis([-1 17 -1 6]); axis equal triplot(DT)

This triangulation cannot be used to represent the domain of the polygon because some triangles cut across the boundary. You need to impose a constraint on the edges that are cut by triangulation edges. Since all edges have to be respected, you need to constrain all edges. The steps below show how to constrain all the edges.

Enter the constrained edge definition. Observe from the annotated figure where you need constraints (between (`V1`

, `V2`

), (`V2`

, `V3`

), and so on).

C = [1 2; 2 3; 3 4; 4 5; 5 6; 6 7; 7 8; 8 1];

In general, if you have `N`

points in a sequence that define a polygonal boundary, the constraints can be expressed as `C = [(1:(N-1))' (2:N)'; N 1];`

.

Specify the constraints when you create the `delaunayTriangulation`

.

DT = delaunayTriangulation(P,C);

Alternatively, you can impose constraints on an existing triangulation by setting the `Constraints`

property: `DT.Constraints = C;`

.

Plot the triangulation and polygon.

figure('Color','white') subplot(2,1,1); axis([-1 17 -1 6]); axis equal triplot(DT) hold on; patch(P(:,1),P(:,2),'-r','LineWidth',2, ... 'FaceColor','none','EdgeColor','r'); hold off % Plot the standalone triangulation in a subplot. subplot(2,1,2); axis([-1 17 -1 6]); axis equal triplot(DT)

The plot shows that the edges of the triangulation respect the boundary of the polygon. However, the triangulation fills the concavities. What is needed is a triangulation that represents the polygonal domain. You can extract the triangles within the polygon using the `delaunayTriangulation`

method, `isInterior`

. This method returns a logical array whose `true`

and `false`

values that indicate whether the triangles are inside a bounded geometric domain. The analysis is based on the Jordan Curve theorem, and the boundaries are defined by the edge constraints. The ith triangle in the triangulation is considered to be inside the domain if the ith logical flag is true, otherwise it is outside.

Now use the `isInterior`

method to compute and plot the set of domain triangles.

% Plot the constrained edges in red. figure('Color','white') subplot(2,1,1); plot(P(C'),P(C'+size(P,1)),'-r','LineWidth', 2); axis([-1 17 -1 6]); % Compute the in/out status. IO = isInterior(DT); subplot(2,1,2); hold on; axis([-1 17 -1 6]); % Use triplot to plot the triangles that are inside. % Uses logical indexing and dt(i,j) shorthand % format to access the triangulation. triplot(DT(IO, :),DT.Points(:,1), DT.Points(:,2),'LineWidth', 2) hold off;

The Delaunay algorithms in MATLAB construct a triangulation
from a unique set of points. If the points passed to the triangulation
function, or class, are not unique, the duplicate locations are detected
and the duplicate point is ignored. This produces a triangulation
that does not reference some points in the original input, namely
the duplicate points. When you work with the `delaunay`

and `delaunayn`

functions, the presence of
duplicates may be of little consequence. However, since many of the
queries provided by the `delaunayTriangulation`

class are index based,
it is important to understand that `delaunayTriangulation`

triangulates
and works with the unique data set. Therefore, indexing based on the
unique point set is the convention. This data is maintained by the `Points`

property
of `delaunayTriangulation`

.

The following example illustrates the importance of referencing
the unique data set stored within the `Points`

property
when working with `delaunayTriangulation`

:

```
P = gallery('uniformdata',[25 2],0);
P(18,:) = P(8,:)
P(16,:) = P(6,:)
P(12,:) = P(2,:)
DT = delaunayTriangulation(P)
```

`Points`

property shows that the
duplicate points have been removed from the data.DT = delaunayTriangulation with properties: Points: [22x2 double] ConnectivityList: [31x3 double] Constraints: []

`DT.Points`

.
Therefore, use the following code to compute and plot the convex hull:K = DT.convexHull(); plot(DT.Points(:,1),DT.Points(:,2),'.'); hold on plot(DT.Points(K,1),DT.Points(K,2),'-r');

`delaunayTriangulation`

,
then the result would be incorrect. The `delaunayTriangulation`

works
with indices that are based on the unique data set `DT.Points`

.
For example, the following would produce an incorrect plot, because `K`

is
indexed with respect to `DT.Points`

and not `P`

:K = DT.convexHull(); plot(P(:,1),P(:,2),'.'); hold on plot(P(K,1),P(K,2),'-r');

`delaunayTriangulation`

.
Doing this eliminates the potential for confusion. This can be accomplished
using the `unique`

function as
follows: P = gallery('uniformdata',[25 2],0); P(18,:) = P(8,:) P(16,:) = P(6,:) P(12,:) = P(2,:) [~, I, ~] = unique(P,'first','rows'); I = sort(I); P = P(I,:); DT = delaunayTriangulation(P) % The point set is unique